Small game hunting
For small game hunting, the usual cartridges are the rimfire .22 LR and .22 WMR (Magnum). Over most of the country small game consists of squirrels and rabbits. In most states varmints and the smaller predators are also legal handgun fare, and there is no closed season. This usually includes such animals as gophers, ground squirrels, starlings, rats, foxes, coyotes, marmots, and jackrabbits. The .22 WMR is adequate for all of these, and in good hands the .22 LR will take many of them.

At handgun ranges, the .22 LR is the queen of small game hunting cartridges. Loaded with high or hyper velocity hollow point cartridges, the .22 LR can humanely and efficiently harvest small game the size of rabbits and squirrels. Where legal, many game birds also make fine (and very difficult) targets for the .22 pistolero who insists on taking only head shots. The .22 high-speed hollow point cartridge shoots flat enough to make 75 yard shots, if the shooter and the pistol are up to it.

The small game gun, and the shooter behind it, must be able to reliably put a bullet into a 3" circle, or a 1.5" circle if a head shot is called for, at whatever range small game is engaged. It is the pistol's accuracy or the shooter's ability, rather than the trajectory of either the .22 LR or .22 WMR cartridge, that usually limits effective range for the small game handgun hunter.

An accurate target type revolver or auto pistol is the probably best handgun for small game hunting. New or used, the Colt Diamondback and Smith & Wesson .22 Masterpiece double action revolvers, and the Ruger Super Single Six single action revolver are the wheel guns most commonly recommended for small game hunting. Equally useful are the some of the target or hunting style .22 autoloaders from Ruger, Browning, High Standard, SIG, and Colt. (The Colt and High Standard .22 autos have been in and out of production over the years, but are fairly common on the used market.) All of them should be capable of shooting, from a bench rest, into 1.5" from 25 yards or 3" at 50 yards, and sometimes they will do quite a bit better.
http://www.chuckhawks.com/handgun_hunting.htm